operations research and enterprise systems

Methodologies and Technologies Distributionally Robust Optimization for Scheduling Problem in Call Centers with Uncertain Forecasts.. Keywords: Distributionally robust optimization·Stoch

Dominique de Werra

Greg H Parlier

4th International Conference, ICORES 2015

Lisbon, Portugal, January 10–12, 2015

Revised Selected Papers

Operations Research

and Enterprise Systems

Communications in Computer and Information Science 577

Commenced Publication in 2007

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Alfredo Cuzzocrea, DominikŚlęzak, and Xiaokang Yang

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Simone Diniz Junqueira Barbosa

Pontifical Catholic University of Rio de Janeiro (PUC-Rio),

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St Petersburg Institute for Informatics and Automation of the Russian

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More information about this series at http://www.springer.com/series/7899

Dominique de Werra • Greg H Parlier

Operations Research

and Enterprise Systems

4th International Conference, ICORES 2015

Revised Selected Papers

123

ISSN 1865-0929 ISSN 1865-0937 (electronic)

Communications in Computer and Information Science

ISBN 978-3-319-27679-3 ISBN 978-3-319-27680-9 (eBook)

DOI 10.1007/978-3-319-27680-9

Library of Congress Control Number: 2015956372

© Springer International Publishing Switzerland 2015

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The registered company is Springer International Publishing AG Switzerland

This book includes extended and revised versions of selected papers presented duringthe 5th International Conference on Operations Research and Enterprise Systems(ICORES 2015), held in Lisbon, Portugal, during January 10–12, 2015 ICORES 2015was sponsored by the Institute for Systems and Technologies of Information, Controland Communication (INSTICC) and co-sponsored by the Portuguese Association ofOperational Research (Apdio)

The purpose of the International Conference on Operations Research and EnterpriseSystems is to bring together researchers, engineers, and practitioners interested in bothresearch and practical applications in the field of operations research Two simulta-neous tracks were held, one focused on methodologies and technologies and the other

on practical applications in specific areas

ICORES 2015 received 89 paper submissions from 38 countries across six nents Of these, 21 % were presented at the conference as full papers These authorswere then invited to submit extended versions of their papers Each submission wasevaluated during a double-blind review by the conference Program Committee Thebest 18 papers were selected for publication in this book

conti-ICORES 2015 also included four plenary keynote lectures from internationally tinguished researchers: Francisco Ruiz, (University of Málaga, Spain), Marc Demange(School of Mathematical and Geospatial Sciences, RMIT University, Australia), MarinoWidmer (University of Fribourg, Switzerland), and Bernard Ries (UniversitéParis-Dauphine, France) We gratefully acknowledge their invaluable contribution asrenowned experts in their respective areas They presented cutting-edge work, thusenriching the scientific content of the conference

dis-We especially thank all authors whose research and development efforts arerecorded here The knowledge and diligence of our reviewers were also essential toensure that high-quality papers were presented at the conference and published herein.Finally, our special thanks to all members of the INSTICC team for their indispensableadministrative skills and professionalism, both of which contributed to awell-organized, productive, and memorable conference

Greg H Parlier

Program Committee

of Sciences), Russian FederationEl-Houssaine Aghezzaf Ghent University, Belgium

Jean-Charles Billaut Ecole Polytechnique de l’Université François-Rabelais

de Tours, France

Spain

Universidade do Minho, Portugal

Xavier Delorme Ecole Nationale Supérieure des Mines de Saint-Etienne,

France

Clarisse Dhaenens French National Institute for Research in Computer

Science and Control, France

Nikolai Dokuchaev Curtin University, Australia

Christophe Duhamel Université Blaise Pascal, Clermont-Ferrand, France

Gintautas Dzemyda Vilnius University, Lithuania

Muhammad Marwan

Muhammad Fuad

University of Tromsø, Norway

Juan José Salazar

Gonzalez

Universidad de La Laguna, SpainChristelle Guéret University of Angers, France

Joanna Józefowska Poznan University of Technology, Poland

Philippe Lacomme Université Clermont-Ferrand 2, Blaise Pascal, France

SAR China

Helena Ramalhinho

Lourenço

Universitat Pompeu Fabra, Spain

Concepción Maroto Universidad Politécnica de Valencia, Spain

Pedro Coimbra Martins Polytechnic Institute of Coimbra, Portugal

VIII Organization

Carlo Meloni Politecnico di Bari, Italy

Jairo R Montoya-Torres Universidad de La Sabana, Colombia

Mohammad

Oskoorouchi

California State University-San Marcos, USA

USA

Marcello Sanguineti University of Genoa, Italy

Dominique de Werra École Polytechnique Fédérale de Lausanne (EPFL),

Switzerland

Gerhard Woeginger Eindhoven University of Technology, The Netherlands

Konstantinos Zografos Lancaster University Management School, UK

Organization IX

Additional Reviewer

Invited Speakers

RMIT University, Australia

X Organization

Methodologies and Technologies

Distributionally Robust Optimization for Scheduling Problem in Call

Centers with Uncertain Forecasts 3Mathilde Excoffier, Céline Gicquel, Oualid Jouini, and Abdel Lisser

A Comparison of a Global Approach and a Decomposition Method for

Frequency Assignment in Multibeam Satellite Systems 21Jean-Thomas Camino, Christian Artigues, Laurent Houssin,

and Stéphane Mourgues

Selection-Based Approach to Cooperative Interval Games 40Jan Bok and Milan Hladík

Re-aggregation Heuristic for Large P-median Problems 54Matej Cebecauer andLˇuboš Buzna

Meeting Locations in Real-Time Ridesharing Problem:

A Buckets Approach 71

K Aissat and A Oulamara

Stochastic Semidefinite Optimization Using Sampling Methods 93Chuan Xu, Jianqiang Cheng, and Abdel Lisser

Evaluation of Partner Companies Based on Fuzzy Inference System

for Establishing Virtual Enterprise Consortium 104Shahrzad Nikghadam, Bahram LotfiSadigh, Ahmet Murat Ozbayoglu,

Hakki Ozgur Unver, and Sadik Engin Kilic

Sara V Rodriguez, and E Juventino Treviño

The Non-Emergency Patient Transport Modelled as a Team Orienteering

Problem 147José A Oliveira, João Ferreira, Luis Dias, Manuel Figueiredo,

and Guilherme Pereira

A Simulation Study of Evaluation Heuristics for Tug Fleet Optimisation

Algorithms 165Robin T Bye and Hans Georg Schaathun

Extended Decomposition for Mixed Integer Programming to Solve a

Workforce Scheduling and Routing Problem 191Wasakorn Laesanklang, Rodrigo Lankaites Pinheiro,

Haneen Algethami, and Dario Landa-Silva

Local Search Based Metaheuristics for Two-Echelon Distribution Network

with Perishable Products 212Sona Kande, Christian Prins, Lucile Belgacem, and Benjamin Redon

Critical Activity Analysis in Precedence Diagram Method Scheduling

Network 232Salman Ali Nisar and Koji Suzuki

Author Index 249

XII Contents

Methodologies and Technologies

Distributionally Robust Optimization

for Scheduling Problem in Call Centers

with Uncertain Forecasts

Mathilde Excoffier(B), C´eline Gicquel, Oualid Jouini, and Abdel Lisser

Laboratoire de Recherche en Informatique - LRI, 91405 Orsay Cedex, France

mathilde.excoffier@lri.fr

Abstract This paper deals with the staffing and scheduling problem in

call centers We consider that the call arrival rates are subject to tainty and are following independent unknown continuous probability dis-tributions We assume that we only know the first and second moments

uncer-of the distribution and thus propose to model this stochastic tion problem as a distributionally robust program with joint chance con-straints Moreover, the risk level is dynamically shared throughout theentire scheduling horizon during the optimization process We propose adeterministic equivalent of the problem and solve linear approximations ofthe Right-Hand Side of the program to provide upper and lower bounds

optimiza-of the optimal solution We applied our approach on a real-life instanceand give numerical results Finally, we showed the practical interest ofthis approach compared to a stochastic approach in which the choice ofthe distribution is incorrect

Keywords: Distributionally robust optimization·Stochastic ming · Joint chance constraints · Mixed-integer linear programming ·

program-Staffing·Shift-scheduling·Call centers·Queuing systems

Practically, scheduling call centers consists in deciding how many agents dling the phone calls should be assigned to work in the forthcoming days orweeks The goal is to minimize the manpower cost while respecting a chosen

han-c

 Springer International Publishing Switzerland 2015

D de Werra et al (Eds.): ICORES 2015, CCIS 577, pp 3–20, 2015.

4 M Excoffier et al.

Quality of Service (QoS) In call centers, we usually consider the expected ing time before being served, or the expected number of clients hanging upbefore being served, i.e the abandonment rate, as a relevant measure of Quality

wait-of Service

The standard model for this problem is based on forecasts of expected callarrival rates These forecasts are computed from historical data giving the num-bers of calls for the working time horizon Since the quantity of calls vary strongly

in time, the working horizon is split in small periods of time, usually 30-minuteperiods Thus we obtain for each period an expected call arrival rate Then weare able to compute the staff requirements for each period from the forecasts and

an objective service level which represents the chosen Quality of Service Thiscomputation is done with the well-known Erlang C model Finally, the numbers

of agents required for the whole working horizon are determined through anoptimization program, using the previous period-by-period results

The shift-scheduling problem presents some characteristics: first, we need tosplit the horizon into small periods of time in order to be able to represent thevariation of rate with the best precision possible This leads to an increasingnumber of variables Second, since we are considering human agents we have torespect several manpower constraints Thus, agents have to follow establishedshifts and can not work only for a few hours Moreover, the solution of theproblem represents humans, so it has to be integers Finally, call arrival ratesare forecasts and thus subject to uncertainty Thus, the final numbers of agentscomputed is subject to uncertainty as well This should be considered in order

to propose a valid model

Typical call centers models consider a queuing system for which the arrivalprocess is Poisson with known mean arrival rates [6] Since the data of the prob-lem are forecasts of arrival rates, the accuracy of this deterministic approach islimited Indeed, these estimations of mean arrival rates may differ from the real-ity Uncertainty is taken into account in several papers, with various approaches.Several published works consider that input parameters of the optimization pro-gram follow known distributions Some deal with continuous distributions [5],discrete distributions [12] or discretizations of a continuous distribution into sev-eral possible scenarios [11,13] or [7] However it can be difficult to estimate whichdistribution is appropriate [10] for call centers and [4] for general problems con-sider a distributionally robust approach The problem deals with minimizing thefinal cost considering the most unfavorable distribution of a family of distribu-tions whose parameters are the given mean and variance In [10], the χ2statistic

is used to build the class of possibles discrete distributions, with a confidenceset around the estimated values [4] consider the set of radial distributions tocharacterise the uncertainty region, but do not solve the final optimization pro-gram for this set Moreover they do not focus on a specific problem and do notconsider integer variables

In the optimization program, we need to take into account and manage therisk of not respecting the objective service level [11,13] choose to penalize thenon respect of the objective service level with a penalty cost in the objective

Distributionally Robust Optimization for Scheduling Problem 5

function of the optimization program [5,9] use a chance-constrained model, inwhich the constraints are probabilities to be respected with the given risk level.[9] focus on the staffing problem but not the scheduling problem, and consideronly one period of time

The contributions of this paper are the following: first we model our problemwith uncertain mean arrival rates and a joint chance-constrained mixed-integerlinear program This approach corresponds well with the real requirements ofthe scheduling problem in call centers Indeed, forecasts are a useful indication ofwhat can happen in reality but can not be considered as enough This approach

is in contrast with most previous publications whose risk management rely on apenalty cost This penality can be difficult to estimate

Second we consider the risk level on the whole horizon of study instead ofperiod by period with joint chance constraints It enables to control the Quality

of Service on the whole horizon of study, which is a critical benefit Managersdemand to have a weekly vision of the call center, and not only for short periods

of time Moreover we propose a flexible sharing out of the risk through theperiods in order to guarantee minimization of the costs As far as we know,this consideration is only used in [5] for the staffing and scheduling problem incall centers

Finally we focus on a distributionally robust approach, considering that weonly know the first two moments of the continuous probability distributions.Since we do not know in reality what is the adequate distribution, we investigate

a way of solving the problem for unknown distributions Unlike other proposeddistributionally robust approaches ([10] in particular), we consider continuousdistributions instead of discrete distributions This allows to a better represen-tation of the reality Moreover, [10] focus on the uncertainty on the parameters

of a known gamma distribution whereas we focus on the uncertainty of the tribution with known parameters

dis-The rest of the paper is organized as follows In Sect.2we present the lation of the problem At first, we propose the staffing model used for computingthe useful data of the scheduling problem Then we introduce the distributionallyrobust chance-constrained approach In Sect.3we propose computations leading

formu-to the deterministic equivalent of the distributionally robust program We alsopresent the piecewise linear approximations leading to the final programs whosesolutions are lower and upper bounds of the initial optimal solution Section4

gives some numerical results Finally5investigates the importance of the choice

of the distribution and thus the benefit of the distributionally robust approach

6 M Excoffier et al.

working hours and breaks, for lunch for example The problem is then to decidehow many working agents need to be assigned to each shift in the call center inorder to respect a choosen objective service level This computation uses data

of calls arrival rates

As previously explained, since arrival rates vary strongly in time, the horizon

is split into T small periods, typically 15 or 30 min For each small period of time

t, forecasts are computed from historical data of numbers of calls Based on these

forecasts of number of incoming calls, we can compute the agents requirements

at each period of time t.

In that goal we use the Erlang C model, [6] At each period of time t we

consider the call center as a queuing system in stationary state [8] This is a

M t /M/N t queue, where the customer arrival process is Poisson with rate λ and the services times are independent and exponentially distributed with rate μ The number of servers, i.e number of agents of our problem, is denoted by N t

for the period t The queue is assumed to have an infinite capacity, with a First

Come-First Served (FCFS) discipline of service

In our problem we consider the average waiting time as the Quality of Service.The Erlang C model gives the function of Average Speed of Answer (ASA) Thisfunction gives the expected waiting time according to the parameters of the

queue: the service rate μ, the arrival rate λ and the number of servers N The

ASA function is the following (see [6] or [3]):



Note In this relation λ and μ are real numbers whereas N is an integer In the

studied problem, the objective service level is a maximum ASA value We denote

ASA ∗this value As in [5], we will introduce a function of λ, μ and ASA ∗givingthe required number of agents, which will be here considered as a real value

The previous ASA (Average Speed of Answer) function is used in an algorithm

to compute the minimum number of agents required to reach the targeted ASA ∗,

given λ and μ.

The procedure is the following:

– We compute ASA(N, λ, μ) and ASA(N + 1, λ, μ) such that

ASA(N, λ, μ)  ASA ∗ and ASA(N + 1, λ, μ) < ASA ∗

We denote ASA(N, λ, μ) as ASA N,λ

Distributionally Robust Optimization for Scheduling Problem 7

– The real value of N is computed by a linearization in the [ASA N,λ ; ASA N +1,λ]segment The affine function is:

ASA ∗ =(ASA N +1,λ − ASA N,λ)∗ b

+ (N + 1) ∗ ASA N,λ − N ∗ ASA N +1,λ

and b is the real value of required agents we are looking for 

For each period, this algorithm gives us the requirement value b as a function

Finally we are able to compute the number of agents b required to respect the objective service level ASA ∗ when the clients arrive at the rate λ and they are served at the rate μ.

The values of b obtained represent estimations of agents requirements Since

our computed results are subject to uncertainty, we consider that they are in factthe means of random variables of requirements By considering real values ratherthan integers through the previous algorithm, we ensure a better precision in theuncertainty management We assume that these variables are independent

In next section, we present the distributionally robust optimization programfor solving the shift-scheduling problem, considering the agents numbers as ran-dom variables

We consider the following chance-constrained shift-scheduling problem:

the matrix of S shifts of T periods The term a i,j is equal to 1 if agents are

working during period i according to shift j and 0 otherwise The agents vector

x is composed of S variables; x i is the number of agents assigned to the shift i Thus there are T constraints, each for one period of time, and the product Ax represents the number of assigned agents for each of these periods Finally,  is

the risk we allow us to take Then 1−  is the confidence interval.

This program minimizes the manpower cost of working agents while ing the chosen objective service level for the horizon time under the risk level

respect- The objective service level is the value ASA ∗ described in previous section.Thus we want to guarantee a maximum expected waiting time for the clientwhile controlling the costs

8 M Excoffier et al.

The chance constraints approach is chosen in order to deal with randomvariables We want to guarantee that the probability that we staffed enoughagents is higher than the given proportion 1−  Then, our program deals with

joint chance constraints Indeed, instead of considering individual constraintsand one risk level for each period, we set the risk for the whole horizon time

We assume that we do not know exactly what distributions the random

variables b t are following, but we know the means ¯b t and the variances σ t2

We focus here on the distributionally robust approach: we do not know whichdistribution is the correct distribution but we want to optimize our problem forall the possible distributions and thus the most unfavourable distribution with

known expected value and variance We note b ∼ (¯b, σ2) the vector of variables

b t, with means ¯b t and variances σ2

t.Then, we consider the following program:

s.t inf

b∼(¯ b,σ2 P {Ax  b}  1 − 

x ∈ (Z+)S ,  ∈]0; 1]

Since we assume that the random variables are independent, we can split the

constraint into T independent constraints We propose here to dynamically share

out the risk through the periods Indeed, instead of choosing how to share outthe risk through the periods before the optimization process, we decide thatthe proportion for each period will be a variable of the optimization program.This flexibility leads to cheaper solutions and are still satisfactory in term ofrobustness [5]

We introduce the variables y twhich represent the proportion of risk allocated

3 Deterministic Equivalent Problem

Let us focus on the expression of one constraint For a given period t, we have:

inf

b t ∼( ¯ b t ,σ2)P {A t x  b t }  (1 − ) y t (7)

Distributionally Robust Optimization for Scheduling Problem 9

Using [2] (Prop.1), we obtain the following result :

Here is an illustration of the piecewise approximations of function f for:

10 M Excoffier et al.

Fig 1 Piecewise linear approximations of function f.

Piecewise Tangent Approximation We give here a lower bound of f :

s.t.∀t ∈ [[1; T ]], ∀j ∈ [[1; n]],

Distributionally Robust Optimization for Scheduling Problem 11

A t x − b t

σ t  δ l,j y t + α l,j T



t=1

y t= 1

x ∈ (Z+)S ,  ∈]0; 1], ∀t ∈ [[1; T ]], y t ∈]0; 1]

where S is the number of shifts and T the number of periods.

Piecewise Linear Approximation Similarly, we give here an upper bound

of the function with a piecewise linear approximation

Let us choose n points y j ∈]0; 1], j ∈ [[1; n]] be n points such that y1< y2< < y n and interpolate linearly between them

We denote ˆf u,j the piecewise linear approximation between the points y jand

y j+1(the subscriptu stands for upper):

∀j ∈ [[1; n − 1]],

ˆ

f u,j (y) = f (y j)+ y − y j

s.t.∀t ∈ [[1; T ]], ∀j ∈ [[1; n − 1]],

A t x − b t  (δ u,j y t + α u,j )σ t T



t=1

y t= 1

x ∈ (Z+)S ,  ∈]0; 1], ∀t ∈ [[1; T ]], y t ∈]0; 1]

where S is the number of shifts and T the number of periods.

In this section we first proposed a deterministic equivalent to the initial tributionally robust stochastic problem Therefore, the optimal solution of thedeterministic program is the optimal solution of the initial program We had todeal with a mixed-integer nonlinear program Second, we provided close upperand lower bounds of the optimal solution by introducing piecewise tangent andlinear approximations This was possible because of the convexity of the con-straints This led to two mixed-integer linear programs whose number of integerand binary variables are not increased compared to the initial formulation These

Next section gives an example of the method to solve a scheduling problem

up the shifts matrix As we previously said in Sect.2, we can standardize the

service rate μ without loss of generality We consider that all agents have the

same hourly salary, thus the cost of one agent is proportional to the number ofperiods worked

We computed the vectors of scheduled agents x l and x ufor one week with thetwo programs (13) and (16) of the previous section, providing an upper boundand a lower bound of the optimal solution cost We used 17 points for computingthe piecewise tangent and linear approximations We noticed that the order of

magnitude of variables y t is between 10−2 and 10−1, thus we reduced the gapbetween the upper and lower bounds by gathering most of the points aroundthis area

We want to evaluate the quality of our solutions x l and x u To this end wesimulate possible realizations of arrival rates according to different distributionswith the same data as previously We consider different possible distributions:gamma distributions, uniform distribution, Pareto distribution, and variations

of normal distributions (log-normal, folded normal)

We elaborate a scenario as following: for each period of time we simulate acall arrival rate according to one of the given probability distributions Then

we compute the number of effective required agents for each period A scenario

Distributionally Robust Optimization for Scheduling Problem 13

covers requirements for the whole time horizon Finally we compare these values

of requirements with our solutions of the problem (lower solution x l and upper

solution x u) A scenario is considered as violated if at least in one period the

scheduled solution by x u or x lis not enough in comparison of what the realizationrequires

We computed between 100 and 500 scenarios for each probability utions The percentage of violations gives us an idea of the robustness of ourapproach for several chosen distributions The cost of the solutions gives us anidea of the quality of the minimization

In Table1, we give the percentage of violated scenarios for various ranges of

values of means and variances, and risk level The queue parameters μ was set

to 1 as it simply represents a multiplicity factor The first column gives the range

of values of the variances through the day The second column gives the range

of values of the means through the day, following a typical seasonality

The value Cost Gap (CG) of the 5th column is given by the relative differencebetween the cost of the upper bound solution and the cost of the lower bound

solution: CG = c t x u −c t x l

c t x l The last column gives the number of violated scenarios for the lower boundand for the upper bound

In Table1 we can notice that both upper and lower bound solutions respectthe set risk level The variations of the parameters show that the bigger thevariances, the better the model The distributionally robust model deals verywell with increasing of variances We notice that even if we allow 15 % risk, only

a few scenarios are violated when the variances are higher (second and last lines

of Table1) In these cases the call center is over-staffed and the given solutionsseem too conservative But it is important to remember that all the observationsare based on simulations of only a few examples of distributions These very low

percentages only show that if the arrival rates λs follow in reality one of the

studied distributions, it may be over-staffed However the distributionally robustmodel indeed consists in taking all possible distributions with given mean andvariance into account Thus it may be possible to reach the maximum risk levelwith other particular distributions

These results show that our approach is robust, considering the numbers ofviolations never exceed the risk level we set The values of Cost Gap show thatthe two bounds are close enough to propose a very close solution to optimalsolution

We can notice that even if the solutions costs are very close, the number ofviolations is different between the upper solution and the lower solution This isdue to the fact that the distribution of the agents through the different shifts isdifferent according to the programs

Table2 focuses on comparing results for different risk levels The simulationswere made with these parameters:

14 M Excoffier et al.

Table 2 Results for different risk levels.

Parameters Results

upper solution lower solution

The first two columns of Table2gives the chosen parameters Columns 3 and

4 gives the solution costs of the two programs and column 5 gives the Cost Gap.Finally, the two last columns give the number of violated scenarios for the twosolutions

Unsurprisingly, the cost of the solution increases when the risk level decreases.The Cost Gap seems to remain in a small range, even if we notice a small increase

of the gap when the risk is lowered

We can also see an increasing of the cost when ASA ∗(the objective AverageSpeed of Answer) decreases

Like previously, the violations results show that our model respects the initialrisk conditions, for both upper and lower solutions

Figure2 show the values of y t variables through the horizon for the upperbound (in blue) and the lower bound (in green) The red line shows the equaldivision of the risk through the day This figure brings out the interest of dynam-

ically sharing out the risk: optimization of the variable y tshows their value aredifferent from the simple equal division through the periods Thus our approach

is more complicated but leads to cheaper solutions than a simpler approach withfixed risk levels

Distributionally Robust Optimization for Scheduling Problem 15

Fig 2 Sharing out of the risk through the day.

vs Distributionally Robust Approach

In the previous section we highlighted the robustness of the distributionallyrobust approach We also noticed that the solution computed by our approachmay be overstaffed However this overstaffing may be understandable since themain consideration is that we do not know what is the right distribution Inthis section, we propose to compare our approach with a standard stochasticapproach under the assumption that we know the distribution of the randomvariables What if we model the problem under the assumption of a known dis-tribution but appears to be the wrong one?

In the following programs, we suppose that the agents requirements are dom variables following a known continuous distribution We consider here thenormal distribution with the known means and variances, as previously

ran-We derive the first stochastic program (4) introduced in Sect.2.3 as in [5],without considering the infimum on the chance constraint

However, the same considerations as previously are still valid: the variablesare independent, we consider a dynamic sharing out of the risk and do piecewiselinearizations leading to upper and lower bounds

For easier computation, we standardize the normal distribution and denote

β the standard normal deviate.

β∼N (0,1)((1− ) y t) is known for a probability 1−  0.5 Hence, piecewise

linearizations described in Sect.3.2 can be applied and are guaranteed to giveupper and lower bounds

The lower bound is given by a first-order Taylor series expansion on n given

points for the tangent approximation The resulting program is:

tion is:

s.t.∀t ∈ [[1; T ]], ∀j ∈ [[1; n − 1]], A t x − b t

σ t  δ j ∗ y t + α j T

Distributionally Robust Optimization for Scheduling Problem 17

The coefficients for the upper bound program are

Using the following parameters, we compute a lower solution x l and an upper

solution x u of the problem:

As with the previous Sect.4, we generated scenarios based on various tributions: gamma distributions, uniform distribution, Pareto distribution, andvariations of normal distributions (log-normal, folded normal) For each distrib-ution, we computed 100 scenarios and indicated the percentage of violations ofour solutions These results are seen in Table3

dis-Table 3 Robustness of the stochastic approach for different wrong distributions

Tar-geted risk level is 5 %.

Percentage of violationsDistribution For lower bound For upper bound

Folded normal 5 % 2 %Log-normal 10 % 6 %

Table3represent the percentages of violations of our solutions if the real tributions are the ones indicated The first line shows the quality of the stochaticapproach when the assumption of normal distribution is right

dis-18 M Excoffier et al.

The second line of the table shows that if the right distribution is the gammadistribution, both the lower bound and the upper bound staffing solutions donot give a satisfactory risk management The solutions were computed in order

to respect a risk of 5 % but 7 % to 15 % of scenarios were violated

Thus we can notice that if we wrongly choose the normal distribution instead

of the Gamma or the Log-normal distribution, the quality of the solution is notsatisfactory anymore The other distributions still respects the targeted risk level

on this batch of scenarios

Note We noticed some rare batches where the scenarios generated with the

Pareto distribution did not respect the risk level for both upper and lower boundssolutions As far as we tested, the Folded normal distribution scenarios showedviolations only for upper bound, but the percentage was high compared to theallowed risk level (between 3 and 4 times)

This result shows that if the choice of the distribution is wrong, the resultedstaffing solutions can not longer be satisfactory This problem does not appear

in our distributionally robust approach, since it is designed intrinsically to dealwith this difficulty Since in some situations it is difficult to guarantee the rightdistribution, the distributionally robust approach is definitely adapted

This paper presents a distributionally robust approach for the staffing and scheduling problem arising in call center We introduced the distributionallyrobust approach, considering that the call arrival rates are following unknowncontinuous distributions Moreover, instead of considering the risk level on aperiod-by-period basis, we decided to set this risk level for the whole horizon ofstudy and thus consider a joint chance-constrained program Then, we proposed

shift-a deterministic equivshift-alent of the distributionshift-ally robust shift-approshift-ach with shift-a dynshift-amicsharing out of the risk We were thus able to propose solutions with reduced costscompared to other published approaches Finally we gave lower and upper bound

of the problem with piecewise linear approximations Computational results showthat both upper and lower solutions respect the objective risk level for a givenset of continuous distributions This shows that our approach proposes robustsolutions The Cost Gap was small enough to be able to bring out a valid solutionfor the initial problem, which is eventually useful for the managers

In the simulations, we noticed that mainly the Pareto distribution andGamma distribution are the ones with violated scenarios The solutions of themodel show that for other distributions, the call center may be over-staffed.Thus, we could study further the call center model in order to evaluate what arethe interesting distributions to consider This can lead, as an improvment forour work in the future, to the study of a given set of distributions, according tosome conditions (in addition to the known mean and variance)

The distributionally robust approach showed an advantage compared to astochastic program with a wrong assumption on the distribution Indeed, if theassumption of normal distribution turns out to be incorrect, the staffing solutions

Distributionally Robust Optimization for Scheduling Problem 19

are not satisfactory whereas the distributionally robust approach considers thispossibility per se

Moreover, we can focus on improving the queuing system model by ering another approach of the representation of the service level in order to have

consid-a closer representconsid-ation to reconsid-ality

Another interesting future research would be to conduct a sensitivity analysisthat accounts for the forecast bias

Finally we made the assumption that periods of the day are independent Inreality, we can notice a daily correlation of the periods in a call center: busy peri-ods may appear in an entire busy day and rarely alone Conversely, light periodsshould lead to an entire light day We can then consider that the effective arrivalrates depend on a busyness factor, which represents this level of occupation of

=ln

2

(p)(1 + 2p y)4(1− p y)2

p y

Since every term of the second derivative is positive, we conclude that d dy2f2 is

20 M Excoffier et al.

References

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perspective on operations management research Prod Oper Manage 16, 665–688

(2007)

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3 Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., Zhao,L.: Statistical analysis of a telephone call center: a queueing-science perspective

J Am Stat Assoc 100, 36–50 (2005)

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programs J Optim Theory Appl 130, 1–22 (2006)

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research prospects Manuf Serv Oper Manage 5, 79–141 (2003)

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9 Gurvich, I., Luedtke, J., Tezcan, T.: Staffing call centers with uncertain demand

forecasts: a chance-constrained optimization approach Manage Sci 56, 1093–1115

(2010)

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in call centers with uncertain arrival rates Optim Methods Softw 28, 501–522

(2013)

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non-stationary arrival rate and flexibility OR Spectr 34, 691–721 (2012)

12 Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach forlinear programs with probabilistic constraints In: Fischetti, M., Williamson, D.P.(eds.) IPCO 2007 LNCS, vol 4513, pp 410–423 Springer, Heidelberg (2007)

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centers with global service level agreements Eur J Oper Res 207, 1608–1619

(2010)

A Comparison of a Global Approach and a Decomposition Method for Frequency Assignment in Multibeam Satellite Systems

Jean-Thomas Camino1,2,3(B), Christian Artigues2,3, Laurent Houssin2,4,

and St´ephane Mourgues1

1 Telecommunication Systems Department, Airbus Defence and Space,

Space Systems, 31 Rue des Cosmonautes, 31402 Toulouse, France

{jean-thomas.camino,stephane.mourgues}@astrium.eads.net

2 CNRS, LAAS, 7 Avenue du Colonel Roche, 31400 Toulouse, France

{artigues,houssin}@laas.fr

3 Universit´e de Toulouse, LAAS, 31400 Toulouse, France

4 Universit´e de Toulouse, UPS, LAAS, 31400 Toulouse, France

Abstract As a result of the continually growing demand for

multi-media content and higher throughputs in wireless communication tems, the telecommunication industry has to keep improving the use

sys-of the bandwidth resources This access to the radisys-ofrequency trum is both limited and expensive, which has naturally lead to thedefinition of the generic class of combinatorial optimization problemsknown as “Frequency Assignment Problems” (FAP) In this article, wepresent a new extension of these problems to the case of satellite systemsthat use a multibeam coverage With the models we propose, we makesure that for each frequency plan produced there exists a correspondingsatellite payload architecture that is cost-efficient and decently complex.Two approaches are presented and compared: a global constraint pro-gram that handles all the constraints simultaneously, and a decompo-sition method that involves both constraint programming and integerlinear programming For the latter approach where two subproblems arestudied, we show that one of them can be modeled as a multiprocessorscheduling problem while the other can either be seen as a path-coveringproblem or a multidimensionnal bin-packing problem depending on theassumptions made These analogies are used to prove that both the sub-problems addressed in the decomposition method belong to the category

spec-of NP-hard problems We also show that, for the most common class spec-ofinterference graphs in multibeam satellite systems, the maximal cliquescan all be enumerated in polynomial time and their number is relativelylow, therefore it is perfectly acceptable to rely on them in the schedulingmodel that we derived Our experiments on realistic scenarios show thatthe decomposition method proposed can indeed provide a solution of theproblem when the global CP model does not

Keywords: Frequency assignment ·Multiprocessor scheduling · Pathcover·Linear programming·Constraint programming·Maximal cliquesenumeration

c

 Springer International Publishing Switzerland 2015

D de Werra et al (Eds.): ICORES 2015, CCIS 577, pp 21–39, 2015.

22 J.-T Camino et al.

1 Introduction

A common characteristic of any telecommunication system is that it is width limited, and one of the main challenges for the system engineers is tooptimally use this precious resource Satellite telecommunications systems are noexception to that rule, and this already difficult task is even more complex whenthe specific limitations and needs of the satellite payload are taken into consider-ation Plenty of literature can be found on the problem of assigning frequenciesunder the name of “Frequency Assignment Problems” (FAP) For instance, [1] is

band-a very thorough survey on the models band-and the optimizband-ation methods thband-at hband-avebeen developed over the years to solve the frequency assignment problems thatemerged in a lot of different wireless communications systems The recent liter-ature proposes more and more sophisticated methods to solve the FAP, such asparallel hyperheuristics [12], differential evolution [10], population-based heuris-tics [8,17] or considers more and more realistic variants of the FAP according

to specific problem characteristics [7,9,16] This article aims at presenting newmodels and approaches for this extension of the frequency assignment problem

to multibeam satellite systems, and promising results on realistic scenarios

Fig 1 The uplink (1), the satellite payload (2) and the downlink (3) of the forward

link of a multibeam satellite system

A multibeam satellite system is characterized by a plurality of relativelynarrow beams used to provide coverage to its service area as shown in Fig.1,each beam being the representation of an antenna gain loss threshold for thecorresponding satellite radio source Still in Fig.1, the role of the satellite payload(2) is to receive, downconvert, amplify, and retransmit the signals of the uplink(1) in the different beams of the downlink (3) where the end-users are located

It is assumed that the system bandwidth is divided into identical frequencychannels, the bandwidth of a channel being equal to that of one carrier signal Foreach beam, it is either specified by the operator or assessed in advance how much

A Comparison of a Global Approach and a Decomposition Method 23

bandwidth is needed and therefore how many carriers must be transmitted in it.Assuming that the carrier uplink frequencies are known or treated afterwards,system engineers have to define for each carrier of each beam:

 The frequency channel used in the downlink

 The polarization of the signal in the downlink

 The high power amplifier in the payload that will be amplifying the

corre-sponding uplink carrier

These are the variables of the problem presented in this paper Values must beassigned to them with the goal to minimize the levels of interferences in eachbeam, the number of high power amplifiers needed in the satellite payload, andthe number of hardware needed for the downconversions More precisely, theapproach we have selected is to aim at minimizing the number of high poweramplifiers needed in the satellite payload since they are heavy, expensive, andhighly power-consuming, while we will be using constraints to limit the interfer-ences and the hardware needed for the downconversions to what is acceptable.The rest of the article is structured as follows In Sect.2, the problem con-straints are listed and detailed Then, Sect.3focuses on the different approaches

we have devised to actually model the problem Finally, Sect.4provides mental results and concrete scenario examples, before some concluding remarks

experi-in Sect.5

For the quality of transmission of a signal, the interferences are a determiningfactor and any frequency assignment procedure should try to minimize them.Let us remind that a frequency and a polarization must be assigned to eachcarrier of each beam in the downlink Note that in this work, the isolation of thesignals through the time-dimension is not considered In the end, the frequencyrelated constraints that are taken into account here are the following:

– Polarization Isolation

A perfect radio antenna transmits and receives waves in a particular tion and is insensitive to orthogonally polarized signals [4], meaning that thesame frequency channel can therefore be used twice in the same area with-out risking severe interferences In actual facts, antennas cannot transmit andreceive perfectly in one polarization only, it is always a combination of twoorthogonal polarizations, one of them being predominant To take advantage

polariza-of that property anyway, the choice here has been to consider that two carriers

at the same frequency using orthogonal polarizations are allowed to be mitted in closer zones than two carriers transmitted at the same frequencyand with the same polarization

trans-24 J.-T Camino et al.

– Spatial Isolation

Thanks to antenna gain losses, two carriers can use the same color (frequency

or frequency-polarization couple) as long as the two corresponding beams aresufficiently distant from each other This is often turned into a constraint ofminimum distance between them, leading the very classic binary interference

constraints The resulting representation is a graph G = (B, E) where each vertex b ∈ B corresponds to the zone covered by a beam and each edge e ∈ E

is a link between two zones where it is not allowed to use the same color

– Limit on the Frequency Channel Reuse Values

Defining an upper-bound for these values allows to balance the number oftimes each channel is used, which reduces the hardware needs for frequencyconversions Since two uplink carriers can only share a downconverter in thesatellite payload if they need the same frequency downconversion, it is inter-esting to be able to define the uplink frequencies so as to have as many ofthese situations as possible, and this balance of the frequency reuse factors inthe downlink is advantageous on that regard

A traveling-wave tube (TWT) is a type of high power amplifier for radio quency signals and a widely used technology for satellite telecommunicationpayloads [4] A TWT must be assigned to each carrier of each beam under thefollowing constraints:

fre-– Minimization of the Number of TWT

A TWT is an expensive technology, one should therefore aim at finding adistribution of the carriers in the TWTs that minimizes their number

– Frequency Ranges

The TWTs can have a bandwidth narrower than the overall system width In that case, payload engineers agree with the equipment manufacturer

band-on a limited number of frequency ranges Therefore, the assignment of carriers

to the TWTs must guarantee that the frequency ranges are supported by theavailable equipment

– Carriers Forbidden to Use the Same TWT

Two carriers cannot be amplified by the same TWT if their amplificationrequirements are too different, because of the non-linearity of the TWT Theseincompatibilities are known in advance

– Single use of the Frequency Channels

A TWT cannot amplify two carriers using the same frequency channel

– Limited Number of Carriers per TWT

A TWT is characterized by its output power level That power is shared bythe carriers, therefore the number of carriers per TWT is upper-bounded

– Contiguity of the Frequencies

The payload complexity can be significantly reduced when there are no quency gaps between the carriers in the same TWT The satisfaction of thisparticular constraint is not systematically required, and this explains the twodifferent models we proposed - with and without this constraint - for thetraveling-wave tube assignment problem described further in the article

fre-A Comparison of a Global fre-Approach and a Decomposition Method 25

The first model we derived is a global constraint program (Sect.3.1) that includesall the aforementioned constraints It has been able to provide really interestingsystem solutions on some scenarios, however, when the number of variables is set

to high realistic values, the global CP model fails at providing solutions or ing unfeasibility in reasonable time That is why a decomposition method hasbeen developed, with a subdivision of the problem into a multiprocessor schedul-ing (Sect.3.2) and a path-covering (Sect.3.3) problems The two approaches, thesingle constraint programming model and the combination of the two submodels,are then compared experimentally in Sect.4

The idea to derive a constraint programming model has been motivated by

an analysis of the constraints on the problem variables (frequency, polarization,TWT) that revealed that global constraints could be used to model a large part ofthe problem A global constraint [2] is a set of constraints for which it is preferable

to treat that set of constraints as a whole than to treat all the constraints ofthat conjunction of constraints individually Using global constraints is a way tohave a better view on the structure of the problem, which is then exploited withpowerful filtering algorithms On that regard, a very significant example is theall different constraint [15]

alldifferent(X) that forces all the variables of the array X to be different In the model below,

we also use the global cardinality constraint

global cardinality constr(X, Y, m, M )

that allows to bound the number of times some items appear in a list, X being that list, Y the set of sought values, m the array of minimum number of occur- rences for each sought value, M the array of maximum number of occurrences

for each sought value Finally, the Gecode convexity global constraint

convex(X)

is used to force the integers of an integer set X to be a convex sequence ({1, 2, 3}

is one while{1, 2, 4} is not) These global constraints are implemented in the open

source solver Gecode [11] that we chose to use

An instance of this particular frequency assignment problem is defined by a set of

N B beams, each beam b ∈ B = {1, · · · , N B } being characterized by the number

n b of carriers transmitted in it, leading to an overall number of carriers

is the notation for the set of indices of the carriers of the bth beam Therefore,

note that the C b sets partition the set C = {1, · · · , N C } The system bandwidth

is divided into N F sub-channels indexed by F = {1, · · · , N F } N T TWTs are

available in the payload, and N P orthogonal polarizations are considered

(typ-ically N P = 2), the corresponding index sets being respectively denoted by T

and P Each carrier c ∈ C must be assigned a frequency channel f c ∈ F , a TWT

tc ∈ T and a polarization p c ∈ P These are the problem variables Two graphs

G = (B, E) and G  = (B, E  ) with E  ⊂ E are defined: an edge of E  forbids

the carriers in the two corresponding beams to use the same frequency

chan-nel whatever the polarization, whereas an edge of E only forbids the multiple

use of the same frequency-polarization couple In the following equations, note

that card(X) denotes the cardinality of the set X Here follows the list of the

constraints expressed with these variables:

– For a given beam b such that n b > 1, the n b carriers must be contiguous in

frequency, use the same TWT, and have the same polarization For such b

values, the constraints are:

∀i ∈ {2, · · · , n b }, t ind(b,1)= tind(b,i) (1)

pind(b,1)= pind(b,i) (2)

find(b,i−1) = find(b,i) − 1 (3)– As discussed in Sect.2.1, channel reuse bounds are a tunable parameter in

input used to limit hardware needs for the downconversions Let R min and

R max be the arrays of size N F of these bounds (note that in practice thelower-bound array is set to 0, it is just there to fit the definition of the globalconstraint that use both arrays), then the corresponding is the following:

– The binary interference constraints associated to E can be expressed as follows for all b, b  ∈ B such that b < b  and (b, b )∈ E:

alldifferent(fc + N F(pc − 1) | c ∈ C b ∪ C b ) (5)

– And for E  , for all b, b  ∈ B such that b < b  and (b, b )∈ E :

A Comparison of a Global Approach and a Decomposition Method 27

– The same frequency cannot be used twice by the carriers of a given TWT:

∀t ∈ T, ∀f ∈ F, card(T t ∩ F f)≤ 1 (7)

where Tt ⊂ C and F t ⊂ C respectively are the set of carriers using the TWT

t and the set of carriers using the frequency channel f , these set variables

being linked to the arrays t and f by side channeling constraints that we do

not provide here for the sake of conciseness

– The contiguity in the TWTs Let us denote byF tthe set of frequency channels

used in the TWT t, these set variables being easily defined with channeling

constraints involving the variable arrays f and t Then, the global constraint

convex does exactly what is sought:

– The maximum number of carriers in a given TWT that is upper bounded by

a tunable parameter n:

– The incompatibilities between the carriers that cannot use the same TWT

Let c, c  ∈ C be two carriers forbidden to use the same TWT, then the

corre-sponding constraint is the following:

– The content of the TWTs must be of a given type Let F1⊂ F and F2⊂ F

be two subparts of the system bandwidth such that F1∪ F2= F These two

sets define two types of acceptable frequency contents for the TWTs, which

means that the carriers in a given TWT must either all be in F1 or all be in

F2, which can be expressed as follows:

∀c, c  ∈ C, f c ∈ F \F2 ∧ f c  ∈ F \F1⇒ t c = t c  (11)The objective is the minimization of the number of available TWTs actually

used That number nused is a variable that can be obtained from the array t

with two successive global counting constraints, the first one generating an array

of the number of times each TWT is used, the second counting the number ofnon-zero values in the latter:

The Scheduling Model An analogy with multiprocessor scheduling problems

is possible for the assignment of frequencies and polarizations, that is for the

subproblem that only concerns the variable arrays f, p, and the constraints (2),(3), (4), (5) and (6) That problem, denoted by (S1), is an extension of the

28 J.-T Camino et al.

model proposed in [6] where the frequency assignment is addressed regardless

of the polarizations Each beam b ∈ B is assimilated to a single operation job

whose processing time, expressed in time units, is non-preemptive and equal thenumber of carriers in that beam Note that such a model is only valid becausethe frequencies of the carriers in a same beam are constrained by constraint (8)

to be contiguous, the contiguousness of frequencies corresponding therefore to

the non-preemptiveness of the processing times Each maximal clique of G  isassimilated to a machine with non-overlapping constraints, while each maximal

clique of G is associated to exactly two machines, one for each polarization For each beam/job b ∈ B, C  b denotes the set of machines that correspond to

the cliques of G  that contain b, while C b,1 and C b,2 are the sets of machines

representing the cliques of G containing b that are respectively associated to the

polarizations 1 and 2 For constraint (4), it is assumed that the only restriction

here is an upper-bound on the reuse factor R ∈ N+of the channels (same bound

for each channel), which leads to the definition of M = {m1, · · · , m R } identical

parallel machines Each job b ∈ B requires simultaneously multiple machines.

More precisely, it must be executed on:

– all the machines ofC 

multi-1 ={c 

1,1 , c  1,2 },

C 1,1={c 1,1,1 , c 1,1,2 } and C 1,2={c 1,2,1 , c 1,2,2 }, we have:

– c  1,1 and c  1,2 associated to the cliques/machines{1, 2} and {1, 3} of G 

– c 1,1,1 and c 1,1,2associated to the machines of first polarization for the cliques

{1, 2, 3} and {1, 3, 4} in G

– c 1,2,1 and c 1,2,2 associated to the machines of second polarization for thecliques{1, 2, 3} and {1, 3, 4} in G

– m1the machine in M used by the beam 1

In the example, the two carriers required in beam 1 use the second and thirdfrequency channels and the first kind of polarization With a common deadline

for all the jobs being equal to the number of frequency channels N F (equal to

4 in Fig.2), one can see that solving this scheduling problem is equivalent tosolving the considered subpart of our frequency assignment problem

it is therefore NP-hard

A Comparison of a Global Approach and a Decomposition Method 29

Fig 2 Example of execution of one job on the machines.

Maximal Cliques Enumeration in Multibeam Satellites Interference Graphs As explained in the previous paragraph, one promising direction to

solve efficiently the scheduling part of the frequency assignment problem ered is to use the cliques of the interference graphs It is thus of interest to studythe theoretical and practical complexity of enumerating the maximal cliques Inmultibeam systems, the analysis of their exhaustive enumeration differs depend-ing on the type of graphs considered: regular layouts or random interferencegraphs

consid-Cliques In Regular Layouts A regular layout is an organization of the beams

that provides a continuous coverage of the zone with overlapping beams thatdescribe an hexagonal lattice, as shown in Fig.1 for instance It is a very com-mon choice for the system engineer since the contiguous coverage it provides can

be a crucial specification of the customer, and also, it requires simpler antenna

designs than a non-uniform layout For a beam b ∈ B, let us denote by c b

the position of its center and by Γ (b) the set of its adjacent beams A mon industrial approach for a regular layout with beams of radius r is to have

An important property of the regular interference graphs with the edgesdefined this way is the following:

Proposition: The maximal cliques of the interference graphs corresponding to

the regular patterns in regular layouts can all be enumerated in polynomial time